A goal of magnetic resonance imaging (MRI) is to image the spatial distribution of the transverse magnetization of an object. To do this, the local properties at a point r may be examined. Point r may be isolated in a main magnetic field B0 using gradient magnets. Spins in the object may be excited with a radio frequency (RF) pulse, causing them to enter a higher energy state and then to precess at a known frequency (f) while they release excess energy in returning to their low energy state. This excess energy can be received by an RF receive coil(s). RF receive coils may be spatially non-selective and thus may integrate over an entire volume. The received signal S(t) may be represented by:
      S    ⁡          (      t      )        =            ∫      x                            ⁢                  Mxy        ⁡                  (          x          )                    ⁢              ⅇ                              -            i                    ⁢                                          ⁢          2          ⁢          π          ⁢                                          ⁢                      k            ⁡                          (              t              )                                ⁢          x                    ⁢                          ⁢              ⅆ        x            
where S(t) is the sum of all the signals produced by all the points releasing their energy. However, the local properties at each point r may be of interest. Complicating processing is the fact that data describing local properties for one point may be influenced by data from other points. Looking at the local properties for a point may include sampling the spatial frequency content of an image and then reconstructing an image. Comparing the received data with a reconstructed image facilitates determining how correctly (e.g., accurately) the reconstruction process worked. While MR imaging is described, similar issues may exist with other imaging methods (e.g., computed tomography (CT), positron emission tomography (PET), single photon emission computed tomography (SPECT)).
Image reconstruction is the process of transforming raw data into a spatial image where the spatial image faithfully represents the object from which the raw data was received. Image reconstruction may be viewed as an inverse problem that involves identifying the input to a system based on knowledge about the output of the system. Regularization involves incorporating some expected properties of the input into the calculations for identifying the input. Generally, MR image reconstruction can be seen as a problem involving solving for an image I(x) given a k-space signal s(k), where:
            S      ⁡              (        k        )              =                  ∫                  -          ∞                                      +            ∞                    ⁢                                                    ⁢                        I          ⁡                      (            x            )                          ⁢                  ⅇ                                    -              i                        ⁢                                                  ⁢            2            ⁢            π            ⁢                                                  ⁢            k            ⁢                                                  ⁢            Δ            ⁢                                                  ⁢            kx                          ⁢                                  ⁢                  ⅆ          x                                where      ⁢                          -      N        <=    k    <=          N      .      
If a full k-space was sampled, then there would be a one-to-one relationship between a signal domain and a frequency domain. However, it may be impractical to acquire a full k-space sample. Instead, partial acquisitions may be acquired. Partial acquisitions may also occur in other imaging methods. With partial acquisitions, a theoretically infinite number of images may match the collected k-space sample. Furthermore, the reconstructed image may include artifacts related to under sampling. So, a challenge is to make a good image I(x) with a signal S(k) that represents less than all the available signal where the signal S(k) may be noisy, and to do so in a timely manner.
A technique to improve reconstructed images involves “filling in” the missing data of a partial k-space. Various techniques may be used to provide the missing data. For example, missing locations may be zero-filled where a zero is entered into locations for which no k-space signal is provided. This is generally unsatisfactory. Other techniques may include, for example, conjugate synthesis, Margosian direct method, homodyne demodulation, Cupper's iterative method, iterative projection onto convex sets (POCS), singular value decomposition, and so on. Iterative methods have also been employed with various amounts of success in various contexts. However, these and indeed most reconstruction methods that reconstruct from a partial k-space may suffer from artifacts related to the under-sampling.
Thus, reconstruction methods that attempt to correct for artifacts are of interest. These methods are concerned with finding a “corrected image” that faithfully represents the object being imaged. In these approaches, an image forming process can be seen to be a linear transformation between an ideal image I(x,y) that would perfectly represent the object from which the k-space signal was received and the measured k-space signal S(k,l), where, for example:
      S    ⁢                  ⁢          (              k        ,        l            )        =            ∑              z        =        0                    M        -        1              ⁢                  ⁢                  ∑                  y          =          0                          N          -          1                    ⁢                          ⁢              I        ⁢                                  ⁢                  (                      x            ,            y                    )                ⁢                                  ⁢        A        ⁢                                  ⁢                  (                      x            ,            y            ,            k            ,            l                    )                                    where A(x,y,k,l) depends on ΔB0, the changing B0 field.        Transforming this equation into a relation between a Fourier transformed reconstructed image I′(x′,y′) and the ideal image I(x,y) yields:        
            I      ′        ⁢                  ⁢          (                        x          ′                ,                  y          ′                    )        =            ∑              x        =        0                    M        -        1              ⁢                  ⁢                  ∑                  y          =          0                          N          -          1                    ⁢                          ⁢              I        ⁢                                  ⁢                  (                      x            ,            y                    )                ⁢                                  ⁢        K        ⁢                                  ⁢                  (                      x            ,            y            ,                          x              ′                        ,                          y              ′                                )                                    where K(x,y,x′,y′) depends on an inverse Fourier transform of A.        
It follows therefore that finding a corrected image involves solving a linear system of equations. There are different methods for solving linear systems of equations. Methods include, for example, conjugate gradient (CG), steepest descent (SD), and so on. Thus, one approach to reconstruction can be seen as solving:I′n=KnIn,                where I(x,y) is an ideal image, I′(x′,y′) is a reconstructed image, and K is a matrix that represents a measure of the geometric distortion of an image.        
This can be seen as a linear equation system of the general type:Ax=b                which are well solved using the CG method.        
Using the CG method, one can attempt to minimize:F(x)=1/2xTAx−bTx                to eliminate errore=Ax−b.        
By letting A=K, b=I′ and x=I, the CG method can be used to solve the normal equation:KHKI=KHI′.
The following steps, which represent one example CG method, can be used to solve for KHKI=KHI′.                Establish initial solution I0 as distorted image I′        Compute first residual r0=I′−KI0         Compute first direction ρ0=Kr0         repeat                    Compute Cm=∥KHrm∥22             Compute Dm=∥Kρm∥22             Compute am=cm/dm             Update solution Im+1=Im+amρm             Update residual rm+1=rm−amKρm             Compute Cm=∥KHrm+1∥22/Cm             update direction ρm+1=KHrm+1+cmρm             m=m+1                        until termination condition is met.        
The above method generally describes how an iterative process (e.g., CG) may proceed during image reconstruction. Note that the forward iteration step is unconstrained.